(The Hill) — House Republicans are weighing what kind of national-level abortion ban legislation to pursue if they win the House majority next year, with a 15-week ban or further on the table.

But even as they cheer the Supreme Court overturning the landmark Roe v. Wade abortion rights bill, GOP congressional leaders have made few promises on specific measures they would pursue. Some Republicans advocate leaving abortion restrictions to the states.

House Republicans previously passed the Pain-Capable Unborn Child Protection Act, which would ban abortion after 20 weeks gestation, in 2015 and in 2017.

Rep. Chris Smith (R-N.J.), the sponsor of the bill in this Congress, told CNN that he was considering increasing the limit to a 15-week ban. Asked about a 15-week abortion ban, House Minority Leader Kevin McCarthy (R-Calif.) told CNN that he would support that. 

But at a press conference on the day the Supreme Court released the decision in Dobbs v. Jackson Women’s Health, he did not mention any specific restrictions that Republicans would pursue in a majority.

“We will continue to look wherever we can go to save as many lives as possible,” McCarthy said.

It is unlikely that any abortion ban passed in a GOP-controlled House in the next Congress would become law. Election analysts believe it is unlikely that Republicans would win a filibuster-proof majority in the Senate, and President Biden would almost certainly veto it if such a bill did pass the upper chamber.

But passing an abortion ban in the House would send a message about where the GOP stands on abortion in a post-Roe landscape, and could signal the type of restrictions that Republicans would implement if they do win total control of government.

House Minority Whip Steve Scalise (R-La.) did say in May that Republicans would take up the Born-Alive Abortion Survivors Protection Act on “day one” in a GOP majority.

The bill, led by Rep. Ann Wagner (R-Mo.), would require care to be given to an infant that survives an abortion procedure. Democrats have argued that a 2002 law already guarantees infants’ legal rights.

The Supreme Court’s Dobbs decision means that restrictions into the first trimester of pregnancy could be legal, and many in the GOP conference advocate for total or near-total abortion bans beyond the 20- or 15-week proposal.

The Life at Conception Act led by Rep. Alex Mooney (R-W.Va.) would recognize rights from the “moment of fertilization.” The legislation has 163 GOP cosponsors, and a discharge petition filed by Rep. Bob Good (R-Va.) has 55 signatures, with 218 needed to force a vote on the bill.

The Heartbeat Protection Act from Rep. Mike Kelly (R-Pa.) would ban abortion after cardiac activity can be detected in a fetus, usually around 6 weeks gestation.

In a letter to McCarthy and Scalise last week, the heads of Heritage Action for America, Susan B. Anthony Pro-Life America, CatholicVote, and several other organizations that oppose abortion called for a GOP House majority to call multiple votes on Kelly’s bill and other bills.

“Only federal law can protect unborn babies from states that will continue to allow and even subsidize abortion on demand up until birth. The House of Representatives is best positioned to lead with a robust pro-life agenda beginning now and accelerating in January should you retake the majority,” the letter said.

Some Republicans, though, are wary of rushing toward passing federal restrictions.

Rep. Kevin Hern (R-Okla.) cheered the Dobbs decision when it came out, but says he takes a “true federalist” approach to legislating restrictions on abortion.

“We’ve tipped the power of the last few years too much to Washington, D.C., which our founders warned us about that,” Hern said. “And the real engagement should be at the state level, where citizens can take a day off from work and go petition – truly exercise their First Amendment rights to go petition their government at their state capitals, and their legislators that live in their hometown.”

Senate Minority Leader Mitch McConnell (R-Ky.) has also softened his tone on national abortion restrictions. After the draft Supreme Court decision overturning Roe was leaked, McConnell said that a national abortion ban was “possible.” But speaking to reporters in Kentucky last week, he said that he does not anticipate a national abortion ban becoming law if Republicans take control of the House, and turned the focus back to states. 

“It takes 60 votes in the Senate for either side to prevail on this issue,” McConnell said. “So I think the democratic process on this issue is going to work out at the state level.”

There could be a risk of a national abortion ban backfiring on Republicans electorally. 

Pew Research Center survey conducted from June 27 to July 4 found that 57 percent of adults disapprove of the Supreme Court decision overturning Roe v. Wade, with 43 percent saying they strongly disapprove.

Anti-abortion advocates, while encouraging a federal abortion ban, are not necessarily taking an aggressive posture to demand House Republicans pass a particular bill next year.

“We support whatever bill is the most ambitious in protecting unborn children and moms that can be reached through democratic consensus,”​ ​said Mallory Carroll, vice president of communications at Susan B. Anthony Pro-Life America.

The group supports all of the main anti-abortion bills that Republicans are advocating for in the House, Carroll said, but is “especially eager to see Congress work to pass some kind of gestation limit that protects unborn children within the womb.”